Short Answer

Show that the difference quotient for $f (x) = \cos x$ is given by
role="math" localid="1646431168099" $\frac{f (x + h) - f (x)}{h} = \frac{\cos (x + h) - \cos (x)}{h} = - \sin x \cdot \frac{\sin h}{h} - \cos x \cdot \frac{1 - \cos h}{h}$

The difference quotient for $f (x) = \cos (x)$ is determined by using the Sum formula for the cosine function as

$\frac{f (x + h) - f (x)}{h} = \frac{\cos (x + h) - \cos (x)}{h} = \frac{(\cos x) (\cos h) - (\sin x) (\sin h) - \cos (x)}{h} = - \sin x \cdot \frac{\sin h}{h} - \cos x \cdot \frac{1 - \cos h}{h}$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given data

The given function is

$f (x) = \cos x$

the equation that needs to prove is

$\frac{f (x + h) - f (x)}{h} = \frac{\cos (x + h) - \cos (x)}{h} = - \sin x \cdot \frac{\sin h}{h} - \cos x \cdot \frac{1 - \cos h}{h}$

Step 2. Derivation

Use Sum formula for the cosine function

$\frac{f (x + h) - f (x)}{h} = \frac{\cos (x + h) - \cos (x)}{h} = \frac{(\cos x) (\cos h) - (\sin x) (\sin h) - \cos (x)}{h} = \frac{- (\sin x) (\sin h)}{h} - \frac{\cos (x) - (\cos x) (\cos h)}{h} = - (\sin x) \frac{(\sin h)}{h} - \frac{(\cos x) (1 - (\cos h))}{h} = - \sin x \cdot \frac{\sin h}{h} - \cos x \cdot \frac{1 - \cos h}{h}$

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Short Answer

Show that the difference quotient for $f (x) = \cos x$ is given by
role="math" localid="1646431168099" $\frac{f (x + h) - f (x)}{h} = \frac{\cos (x + h) - \cos (x)}{h} = - \sin x \cdot \frac{\sin h}{h} - \cos x \cdot \frac{1 - \cos h}{h}$

Step by Step Solution

Step 1. Given data

Step 2. Derivation

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Precalculus Enhanced with Graphing Utilities

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Short Answer

Show that the difference quotient for f(x)=cos xis given byrole="math" localid="1646431168099" f(x+h)-f(x)h=cos(x+h)-cos(x)h=-sin x·sin hh-cos x·1-cos hh

Step by Step Solution

Step 1. Given data

Step 2. Derivation

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Show that the difference quotient for $f (x) = \cos x$ is given by
role="math" localid="1646431168099" $\frac{f (x + h) - f (x)}{h} = \frac{\cos (x + h) - \cos (x)}{h} = - \sin x \cdot \frac{\sin h}{h} - \cos x \cdot \frac{1 - \cos h}{h}$